Chapter 11 Optimum Functionally Gradient Materials for Dental Implant Using Simulated Annealing Ali Sadollah and Ardeshir Bahreininejad Additional information is available at the end of the chapter http://dx.doi.org/10.5772/45640 1. Introduction Biomaterials should simultaneously satisfy many requirements and possess properties such as non-toxicity, corrosion resistance, thermal conductivity, strength, fatigue durability, biocompatibility and sometimes aesthetics. A single composition with a uniform structure may not satisfy all such requirements. Natural biomaterials often possess the structure of Functionally Graded Materials (FGMs) which enables them to satisfy these requirements. FGMs provide the structure with which synthetic biomaterials should essentially be formed. The size of biomaterial components is relatively small. In the case of dental applications, the components are generally smaller than 20 mm. This substantially reduces the difficulty of fabricating such materials due to a mismatch in thermal expansion which causes micro crack formation during the cooling cycle. Biomaterials are essential for life and health in certain cases. They have a generally high added value for their size. Thus, biomaterials form one of the most important areas for the application of FGMs. It is an area for which FGMs, at the present time, are sufficiently developed for practical use. The dental implant is used for restoring the function of chewing and biting, and therefore eating, which is the most fundamental activity of human beings required for living. We are living in an era of longer life expectancy and thus, dental care becomes especially important for better quality of life in old age. Implant may be classified to “implant’’ as an artificial bone for medical use and ‘‘dental implant’’ as an artificial tooth for dental use. The specified properties are slightly different depending on their use. The implants in orthopaedics are used mostly as structurally enforced artificial bone which is inserted inside the corpus. Medical implants lay more weight on strength, toughness, torque in mechanical properties and the specific problem of tribology and abrasion resistance in artificial joint. Dental implant is usually much smaller and is used to reconstruct the masticatory function when tooth root is completely lost or extracted. © 2012 Sadollah and Bahreininejad, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 218 Simulated Annealing – Single and Multiple Objective Problems Implant is placed in the jawbone in the manner to penetrate from the inside to the outside of the bone. The function is quite different at the inside of bone, outside and at the boundary. In the inside of jaw bone, bone affinity and stress relaxation are important and in the outside of bone, that is, in oral cavity, the sufficient strength is necessary. In the application of human body implant, FGM is usually composed of Collagen Hydroxyapatite (HAP) and titanium (Watari et al., 2004; Hedia, 2005). HAP is indeed a principal component in human bones and related tissues. HAP inclusion in forming the dental implant material can bring about an enhanced biocompatibility with the native hosting tissues. The main advantages of using FGM dental implant are: 1) reduction of stress shielding effect on the surrounding bones that usually arises in the presence of fully metallic implants (Hedia, 2005), 2) improvement of biocompatibility with bone tissues (Watari et al., 2004), 3) preventing the thermal-mechanical failure at the interface of HAP coated metallic implants (Wang et al., 2007) and 4) meeting the biomechanical requirements at each region of the bone while enhance the bone remodeling, hereby maintaining the bone’s health status (Yang & Xiang, 2007). The latter is more related to volume fraction of FGM. The first three aspects of using FGM implant have been investigated in the previous studies (Watari et al., 2004; Hedia, 2005; Wang et al., 2007). However, limited knowledge has been available in the effect on bone remodeling due to the use of FGM dental implants. The other issue needs to be systematically studied is how to devise an optimal FGM pattern for dental implant application. It has been widely accepted that a mating mechanical property to the host bone should be made in order to avoid stress shielding (Hedia & Mahmoud, 2004; Hedia et al., 2006) and promote osseointegration and bone remodeling (Chu et al., 2006; Yang & Xiang, 2007). However, there are few reports available which examine whether or not a mating property could result in the best remodeling consequence and ensure a long-term success. Recently, optimization of FGM dental implant was studied by Lin et al. (2009) using the Response Surface Methodology (RSM) and the results show the incompatibility of properties with each other and the need for using multi-objective algorithms to overcome the problem. Another issue concerns the existing material engineering technology which may not allow us to make such mating pattern for individuals in a cost efficient way. As a result, how to optimally tailor FGM pattern for remodeling is of noteworthy implication in developing FGM implantation. This chapter aims at extending a more realistic FGM design for dental implantation. Using Simulated Annealing (SA), the multi-objective optimization model was developed to optimize FGM gradient pattern for desirable on-going bone turnover outcome and mechanical responses. SA algorithm has shown great potential for solving optimization problems as they conduct global stochastic search. The multi-objective optimization problem was solved using SA and the results were compared with the RSM. 2. Properties of FGM dental implant In this study, the configuration of FGM dental implant follows the patterns from literature (Wang et al., 2007; Yang & Xiang, 2007). The material gradient is governed by a power law Optimum Functionally Gradient Materials for Dental Implant Using Simulated Annealing 219 with parameter m, as in Equation (1). The volume fractions of the two-phase composite FGM dental implant can be calculated from the following equations (Hedia, 2005; Wang et al., 2007; Yang & Xiang, 2007): VC=yhm (1) V =(1−V ) (2) m C where Vc denotes the volumetric fraction of HAP/Col (ceramic), Vm denotes the volumetric fraction of titanium (metal), m is a constant to define the variation in material composition, y is the vertical position within the implant region and h is the total length of the implant. Fig. 1 shows the schematic view of FGM dental implant with graded material composition used in dentistry. Figure 1. Schematic view of FGM dental implant with graded material composition. Accordingly, the Young modulus and Poisson ratio can be calculated as (Hedia, 2005):   E =E  Ec+(Em−Ec)vm2/3  (3) 0 cEc+(Em−Ec)(vm23−vm) v=v V +vV (4) m m c c where E0 is the equivalent Young modulus at different regions of the implant, Ec is the Young modulus of HAP/Col, Em is the Young modulus of titanium. vc and vm are the Poisson ratios for HAP/Col and titanium, respectively. The HAP/Col and titanium compositions vary according to the relative length of y/h, with respect to the material gradient m, meaning that m governs the variation in the volumetric fraction of the titanium to HAP/Col compositions. Referring to the properties of FGM implant, the values of Ec and Em are kept 220 Simulated Annealing – Single and Multiple Objective Problems within the range of Ec >>1 GPa and Em >> 110 GPa, respectively (Hedia, 2005; Wang et al., 2007; Yang & Xiang, 2007). Fig. 2 demonstrates the variation of mechanical properties including Young’s modulus and Poisson’s ration for diverse FGM pattern. From Fig. 2, the horizontal axis is the vertical position (y) along FGM dental implant, which is varied from 0 to 10 mm. By observing Fig. 2, y=0 mm indicates the region directly connected to the crown, where FGM has the richest content of titanium when m=10, while the highest content of collagen HAP is obtained when m=0.1. In other words, m=10 and m=0.1, respectively, give the highest and lowest gradients in the Young modulus and Poisson ratio in the region of the crown’s end. Therefore, altering m enables us to tailor the property gradient, thereby providing a means to optimizing the remodeling performance induced by the FGM dental implant. Figure 2. Variation in material properties for different FGM configuration: (a) Young’s modulus, (b) Poisson’s ration. 3. Bone remodeling calculations The biomechanical environment changes considerably when using FGM dental implant. Consequently, the bone remodels itself to adapt to the new changes that is imposed on it by minimizing the difference between the new mechanical response and related equilibrium state. Strain energy density is one of the most important mechanical stimuli to explain the bone remodeling (Weinans et al., 1992). The mathematical equations of bone remodeling are Optimum Functionally Gradient Materials for Dental Implant Using Simulated Annealing 221 given as follows (Huiskes et al., 1987; Weinans, 1992; Turner et al., 1997; Turner, 1998; Lin et al., 2008a, 2008b): Bone apposition: ∂ρ η η =B( −(1+ξ)Ξ) if (1+ξ)Ξ (5) ∂t ρ ρ Bone equilibrium: ∂ρ η =0 if(1−ξ)Ξ≤ ≤(1+ξ)Ξ (6) ∂t ρ Bone resorption: ∂ρ η η =B( −(1−ξ)Ξ) if (1−ξ)Ξ (7) ∂t ρ ρ where η denotes the mechanical stimulus (i.e. strain energy density), ρ is the bone density, B is the remodeling constant set to 1gr/cm3 (Weinans et al., 1992), Ξ is the remodeling reference value equal to 0.004 J/g (Weinans et al., 1992; Turner, 1998), and ξ is the bandwidth of bone remodeling with an adapted value of 10% (Weinans et al., 1992). After the bone density values are calculated from the remodeling equations, the Young modulus of cortical and cancellous bones (in GPa) can be updated by using the following equations (Rho et al., 1995; O’Mahony et al., 2001): E =−23.93+24ρ (8) cortical cortical E =2.349ρ2.15 (9) cancellous cancellous Equations (8) and (9) are utilized to update Young modulus after the densities are determined via the remodeling calculations. The internal bone remodeling system is formed using Equation (5) to Equation (9). 4. Design optimization problem The bone remodeling provides quantitative data of changes in bone densities and the stiffness of dental apparatus. The former indicates how the bones react to the change in biomechanical environment in terms of the variation in bone morphology. The latter indicates how the bone remodeling alters the mechanical response, thereby stabilizing the implant and in turn strengthening the bone. In this research, the changes in bone densities and vertical displacement are taken as the direct measures of on-going performance of implantation. From the biomechanical point of view, increase in surrounding bone density and decrease in the occlusal displacement indicate the positive sign to a long-term success in dental 222 Simulated Annealing – Single and Multiple Objective Problems implantation. Thus, design of FGM gradient parameter (m) is expected to maximize the densities and minimize the displacement, which in a form of multi-objective optimization may be represented as: minf(m)= 1  1 D cortical  minf (m)= 1 (10)  2 D cancellous minf3(m)=u(m) subject to 0.1≤m≤10 where f1, f2 and f3 represent the objective functions, Dcortical and Dcancellous are the densities of cortical and cancellous bones, respectively and u(m) denotes the vertical displacement at the top of artificial crown. The objective functions f1, f2 and f3 represent the condition of FGM dental implant at month 48 (Lin et al., 2009). These polynomial response functions are given as: D (m)=−2e−6m6+8e−5m5−1e−3m4+6.2e−3m3−1.9e−2m2+1.76e−2m+1.9297 (11) cortical D (m)=2e−6m6−6e−5m5+6e−4m4−2.6e−3m3+3e−3m2−2.4e−3m+1.1712 (12) cancellous u(m)=7e−10m6−2e−8m5−3e−7m4−2e−6m3+7e−6m2−1e−5m+4e−5 (13) The polynomial response functions were obtained using experimental tests on various quantities of material gradient on the FGM dental implant. Lin et al. (2009) proposed to adopt the RSM to construct the appropriate objective functions for a single design variable problem. Fig. 3 represents the cortical and cancellous densities, and displacement versus material gradient (m). As m increases, the displacement function u(m) is increased as shown in Fig. 3c. The increase in m results in an increase in the cortical density as shown in Fig. 3a. From Fig. 3b, the increase in m results the decrease in the density of cancellous. The design objectives were to maximize both Dcortical and Dcancellous in order to determine the best possible material configuration for implant that will give the maximum amount of bone remodeling. At the same time, the objective function f3 is minimized in order to reduce the downward implant displacement. An ideal situation would be to attain a consistent optimal material gradient, where the maximum bone turnover can be made and the displacement is kept to minimal. To explore such a multi-objective design, first, the two objective optimization problems of either cortical or cancellous densities versus the displacement were formulated as follows: minf = 1  1 D cortical  minf =max(u(m)) (14) 3 subject to 0.1≤m≤10   and Optimum Functionally Gradient Materials for Dental Implant Using Simulated Annealing 223 minf = 1  2 D cancellous  minf =max(u(m)) (15) 3 subject to 0.1≤m≤10   Figure 3. Mass densities and displacement for FGM dental implant: (a) cortical density, (b) cancellous density, (c) displacement (u) (The horizontal axis is m). Fig. 4 shows the trend of cortical density function (f1), cancellous density function (f2) and the displacement function (f3). From Figs. 4b and 4c, it is clear that f2 and f3 have the same trend and behavior and it is expected to obtain one optimal solution. Based on Fig. 3c, we need to minimize the maximum of displacement in order to obtain the minimum value of displacement. The function f3 is minimized by multiplying u(m) by -1. In other hand, Figs. 4a and 4c show the different trend and behavior which with increasing m, the cortical density function decreases while, the displacement function increase. Therefore, it is anticipated to obtain a range of optimal solutions. The problem investigated in this chapter, is taken from (Lin et al., 2009) where the results were obtained using RSM. 224 Simulated Annealing – Single and Multiple Objective Problems Figure 4. Mass densities and displacement functions for FGM dental implant: (a) cortical density function (f1), (b) cancellous density function (f2), (c) displacement function (f3) (The horizontal axis is m). 5. Multi-objective optimization The multi-objective optimization has become an important research topic for scientists and researchers. This is mainly due to the multi-objective nature of real life problems. It is difficult to compare results of multi-objective methods to single objective techniques, as there is not a unique optimum in multi-objective optimization as in single objective optimization. Therefore, the best solution in multi-objective terms may need to be decided by the decision maker. Optimum Functionally Gradient Materials for Dental Implant Using Simulated Annealing 225 The increasing acceptance of SA for solving multi-objective optimization problems is due to their ability to: 1) find multiple solutions in a single run, 2) work without derivatives, 3) converge speedily to Pareto-optimal solutions with a high degree of accuracy, 4) handle both continuous functions and combinatorial optimization problems with ease and 5) are less susceptible to the shape or continuity of the Pareto front (Suman & Kumar, 2006). 5.1. Pareto-optimal solutions The concept of the Pareto-optimal solutions was formulated by Vilfredo Pareto in the 19th century (Rouge, 1896). Real life problems require simultaneous optimization of several incommensurable and often conflicting objectives. Usually, there is no single optimal solution. However, there may be a set of alternative solutions. These solutions are optimal in the wider sense that no other solutions in the search space are superior to each other when all the objectives are considered. They are known as Pareto-optimal solutions. When the objectives associated with any pair of non-dominated solutions are compared, it is found that each solution is superior with respect to at least one objective. The set of non-dominated solutions to a multi-objective optimization problem is known as the Pareto-optimal set (Zitzler & Thiele, 1998). 6. Simulated annealing In 1953, Metropolis developed a method for solving optimization problems that mimics the way thermodynamic systems go from one energy level to another (Metropolis et al., 1953). He thought of this after simulating a heat bath on certain chemicals. This method is called Simulated Annealing (SA). Kirkpatrick et al. (1983) originally thought of using SA on a number of problems. The name and inspiration come from annealing in metallurgy, a technique involving heating and controlled cooling of a material to increase the size of its crystals and reduce their defects. The heat causes the atoms to become free from their initial positions (a local minimum of the internal energy) and wander randomly through states of higher energy. The system is cooled and as the temperature is reduced the atoms migrate to more ordered states with lower energy. The final degree of order depends on the temperature cooling rate. The slow cooling process is characterized by a general decrease in the energy level for with occasional increase in energy. On the other hand, a fast cooling process, known as quenching, is characterized by a monotonic decrease in energy to an intermediate state of semi-order which is used as temperature schedule in this chapter. At the final stages of the annealing process, the system’s energy reaches a much lower level than in rapid cooling (quenching). Annealing (slow cooling) therefore allows the system to reach lower global energy minimum than is possible using a quick quenching process, equivalent to a local energy minimum. By analogy with this physical process, each step of the SA algorithm replaces the current solution by a random "nearby" solution, chosen with a probability that depends both on the 226 Simulated Annealing – Single and Multiple Objective Problems difference between the corresponding function values and also on a global parameter T (temperature), that is gradually decreased during the process. The dependency is such that the current solution changes almost randomly when T is large, but increasingly "downhill" as T goes to zero (Fleischer, 1995). The allowance for "uphill" moves potentially saves the method from becoming stuck at local optima. Several parameters need to be included in an implementation of SA. These are summarized by Davidson and Harel (1996): • The set of configurations, or states, of the system, including an initial configuration (which is often chosen at random). • A generation rule for new configurations, which is usually obtained by defining the neighborhood of each configuration and choosing the next configuration randomly from the neighborhood of the current one. • The target, or cost function, to be minimized over the configuration space. (This is the analogue of the energy). • The cooling schedule of the control parameter, including initial values and rules for when and how to change it (This is the analogue of the temperature and its decreases). • The termination condition, which is usually based on the time and the values of the cost function and/or the control parameter. SA is a popular optimization algorithm due to the simplicity of the model and its implementation. However, due to CPU time-consuming nature of standard SA, a fast temperature schedule to fulfill the required conditions is suggested. In fact, simulated annealing algorithm with the fast cooling process is called simulated quenching (SQ) which is used as an optimization method in this chapter to overcome the slow SA optimization process. 7. SA for multi-objective optimization SA has been used as an optimization method for solving a wide range of combinatorial optimization problems. It has also been adapted for solving multi-objective problems due to its simplicity and capability of producing a desirable Pareto set of solutions. In addition, it is not susceptible to the shape of the Pareto set, since shape may be considered as a concern for mathematical programming techniques. 7.1. The method of Suppapitnarm and Parks (SMOSA) The concept of archiving the Pareto-optimal solutions for solving multi-objective problems with SA has been used by Suppapitnarm et al. (2000). The Suppapitnarm Multi-Objective Simulated Annealing (SMOSA) enables the search to restart from an archived solution in a solution region, where each of the pair of non-dominated solutions may be superior with respect to at least one objective. Since SA only generates a single solution at a given iteration, an independent archive is required to record all non-dominated solutions found during search. Pioneering work in this area was first performed by Engrand (1997), and was further developed by Suppapitnarm et al. (2000).